Chiral Gaussian Unitary Ensemble Research Articles

Winding number statistics for chiral random matrices: Averaging ratios of determinants with parametric dependence

Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the chiral Gaussian unitary ensemble to other chiral random matrix ensembles. In particular, we address the two chiral symmetry classes, unitary (AIII) and symplectic (CII), and we analytically compute ensemble averages for ratios of determinants with parametric dependence. To this end, we employ a technique that exhibits reminiscent supersymmetric structures, while we never carry out any map to superspace.

Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the kth and (k + 1)st smallest eigenvalues in the chiral Gaussian unitary ensemble, both for finite- and large-n. The result depends on the number of the generic zero modes ν and the number of flavours N f, which are given in terms of characteristic polynomials, as motivated by quantum chromodynamics (QCD). We find that the convergence in n is very rapid. The same can be said separately about the limit k → ∞ (limit to the bulk) and ν → ∞ (limit to the soft edge). Interestingly, the Wigner surmise is a very good approximation for all these cases and, apart from k = 1, shows a deviation below one percent. These findings are corroborated with Monte-Carlo simulations. We finally compare for k = 1 with data from QCD on the lattice, being in this symmetry class.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

Symmetry crossover protecting chirality in Dirac spectra

We consider a random matrix model in the hard edge limit (local spectral statistics at the origin in the limit of large matrix size) which interpolates between the Gaussian unitary ensemble (GUE) and the chiral Gaussian unitary ensemble (chGUE). We show that this model is equivalent to the low-energy limit of certain QCD-like theories in the epsilon-regime. Moreover, we present a detailed derivation of the microscopic level density as well as the partially quenched and unquenched partition functions. Some of these results have been announced in a former letter by us. Our derivation relies on the supersymmetry method and is performed here step by step. Additionally, we compute the chiral condensate and the pion condensate for the quenched as well as unquenched settings. We also investigate the limits to GUE and chGUE and confirm our conjecture that the non-uniformity of the GUE limit would carry over to the hard edge limit.

GUE-chGUE transition preserving chirality at finite matrix size

We study a random matrix model which interpolates between the singular values of the Gaussian unitary ensemble (GUE) and of the chiral Gaussian unitary ensemble (chGUE). This symmetry crossover is analogous to the one realized by the Hermitian Wilson Dirac operator in lattice QCD, but our model preserves chiral symmetry of chGUE exactly unlike the Hermitian Wilson Dirac operator. This difference has a crucial impact on the statistics of near-zero eigenvalues, though both singular value statistics build a Pfaffian point process. The model in the present work is motivated by the Dirac operator of 3D staggered fermions, 3D QCD at finite isospin chemical potential, and 4D QCD at high temperature. We calculate the spectral statistics at finite matrix dimension. For this purpose, we derive the joint probability density of the singular values, the skew-orthogonal polynomials and the kernels for the k-point correlation functions. The skew-orthogonal polynomials are constructed by the method of mixing bi-orthogonal and skew-orthogonal polynomials, which is an alternative approach to Mehta’s one. We compare our results with Monte Carlo simulations and study the limits to chGUE and GUE. As a side product, we also calculate a new type of a unitary group integral.

Symmetry Transition Preserving Chirality in QCD: A Versatile Random Matrix Model.

We consider a random matrix model which interpolates between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble while preserving chiral symmetry. This ensemble describes flavor symmetry breaking for staggered fermions in 3D QCD as well as in 4D QCD at high temperature or in 3D QCD at a finite isospin chemical potential. Our model is an Osborn-type two-matrix model which is equivalent to the elliptic ensemble but we consider the singular value statistics rather than the complex eigenvalue statistics. We report on exact results for the partition function and the microscopic level density of the Dirac operator in the ϵ regime of QCD. We compare these analytical results with MonteCarlo simulations of the matrix model.

On characteristic polynomials for a generalized chiral random matrix ensemble with a source

We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a random matrix taken from a L-deformed chiral Gaussian Unitary Ensemble with an external source Ω. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Fyodorov (2017 arXiv:1710.04699), is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated complex bulk/chiral edge scaling regime we retrieve the kernel related to Bessel/Macdonald functions.

Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel–Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.

Random matrix theory for the Hermitian Wilson Dirac operator and the chGUE-GUE transition

We introduce a random two-matrix model interpolating between a chiral Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE) and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit) this theory is in one to one correspondence to the partition function of Wilson chiral perturbation theory in the epsilon regime, such as the related two matrix-model previously introduced in refs. [20,21]. For a generic number of flavours and rectangular block matrices in the chGUE part we derive an eigenvalue representation for the partition function displaying a Pfaffian structure. In the quenched case with nu=0,1 we derive all spectral correlations functions in our model for finite-n, given in terms of skew-orthogonal polynomials. The latter are expressed as Gaussian integrals over standard Laguerre polynomials. In the weakly non-chiral microscopic limit this yields all corresponding quenched eigenvalue correlation functions of the Hermitian Wilson operator.

Are Bosonic Replicas Faulty?

Motivated by the ongoing discussion about a seeming asymmetry in the performance of fermionic and bosonic replicas, we present an exact, nonperturbative approach to both fermionic and bosonic zero-dimensional replica field theories belonging to the broadly interpreted beta=2 Dyson symmetry class. We then utilize the formalism developed to demonstrate that the bosonic replicas do correctly reproduce the microscopic spectral density in the QCD-inspired chiral Gaussian unitary ensemble. This disproves the myth that the bosonic replica field theories are intrinsically faulty.

The solution of a chiral random matrix model with complex eigenvalues

We describe in detail the solution of the extension of the chiral Gaussian Unitary Ensemble (chGUE) into the complex plane. The correlation functions of the model are first calculated for a finite number of N complex eigenvalues, where we exploit the existence of orthogonal Laguerre polynomials in the complex plane. When taking the large-N limit we derive new correlation functions in the case of weak and strong non-Hermiticity, thus describing the transition from the chGUE to a generalized Ginibre ensemble. Applications to the Dirac operator eigenvalue spectrum in QCD with non-vanishing chemical potential are briefly discussed. This is an extended version of arXiv:hep-th/0204068.

Replica field theories, painlevé transcendents, and exact correlation functions.

Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painlevé transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitian random matrices. Further applications are briefly discussed.

On correlation functions of characteristic polynomials for chiral Gaussian unitary ensemble

We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a N× N random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding a Itzykson–Zuber type integral for matrices from the non-compact manifold Gl(n, C)/ U(1)×⋯× U(1) (matrix Macdonald function). The correlation function is shown to be always represented in a determinant form generalizing the known expressions for only positive moments. Finally, we present the asymptotic formula for the correlation function in the large matrix size limit.

Spectral properties of a generalized chiral Gaussian unitary ensemble.

We consider a generalized chiral Gaussian Unitary ensemble (chGUE) based on a weak confining potential. We study the spectral correlations close to the origin in the thermodynamic limit. We show that for eigenvalues separated up to the mean level spacing, the spectral correlations coincide with those of chGUE. Beyond this point, the spectrum is described by an oscillating number variance centered around a constant value. We argue that the origin of such a rigid spectrum is due to the breakdown of the translational invariance of the spectral kernel in the bulk of the spectrum. Finally, we compare our results with the ones obtained from a critical chGUE recently reported in the literature. We conclude that our generalized chGUE does not belong to the same class of universality as the above mentioned model.

Spectral correlations of the massive QCD Dirac operator at finite temperature

We use the graded eigenvalue method, a variant of the supersymmetry technique, to compute the universal spectral correlations of the QCD Dirac operator in the presence of massive dynamical quarks. The calculation is done for the chiral Gaussian unitary ensemble of random matrix theory with an arbitrary Hermitian matrix added to the Dirac matrix. This case is of interest for schematic models of OCD at finite temperature.

Microscopic spectrum of the QCD Dirac operator with finite quark masses

We compute the microscopic spectrum of the QCD Dirac operator in the presence of dynamical fermions in the framework of random-matrix theory for the chiral Gaussian unitary ensemble. We obtain results for the microscopic spectral correlators, the microscopic spectral density, and the distribution of the smallest eigenvalue for an arbitrary number of flavors, arbitrary quark masses, and arbitrary topological charge.